Multiple choice

In the right-angled triangle XYZ, in which the hypotenuse XZ has a length of 30 cm and a perpendicular line segment YN is constructed on side XZ, what is the maximum length in centimetres for the line segment YN?

  1. 5

  2. 10

  3. 15

  4. 20
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C Correct answer
Explanation

In a right triangle, the altitude to the hypotenuse is maximized when the triangle is isosceles. In an isosceles right triangle with hypotenuse 30, the altitude to the hypotenuse is half the hypotenuse, which is 15.

AI explanation

For a right-angled triangle, the length of the altitude to the hypotenuse is given by the product of the legs divided by the hypotenuse. By the arithmetic mean-geometric mean inequality, this product is maximized when the two legs are of equal length, making the triangle an isosceles right triangle. In that maximizing case, the hypotenuse of 30 cm relates to the legs by a² + a² = 30², so 2a² = 900 and a = 15√2. The maximum altitude is then (15√2 × 15√2) / 30, which equals 450 / 30 or 15. The maximum length for the line segment YN is 15.

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